Reversibility of entanglement assisted coding

Cite this problem as Problem 17.

Problem

For any two quantum channels S and T, define the ”entanglement-assisted capacity” C_\text{E}(T,S) of T for S-messages as the supremum of all rates r such that, for large n, rn parallel copies of T may be simulated by n copies of S, where the simulation involves arbitrary coding and decoding operations using (if necessary) arbitrarily many entangled pairs between sender and receiver, and where the errors go to zero as n\to\infty.

Show that C_\text{E}(T,S)=C_\text{E}(S,T)^{-1}.

As for other capacities, the ”two-step coding inequality” C_\text{E}(T,S) \,C_\text{E}(S,R)\leq C_\text{E}(T,R) is easy to show. Hence C_\text{E}(T,S)C_\text{E}(S,T)\leq 1. Equality means here, that the two channels are essentially equivalent as a resource for simulating other channels R (apart from a constant factor c): C_\text{E}(R,S)=c C_\text{E}(R,T) (with c =C_\text{E} (T,S)). In this case we call S and T reversible for entanglement-assisted coding.

Background

For ordinary capacity C(T,S) (without entanglement assistance) reversibility fails in general: When S is an ideal classical 1 bit channel, and T is an ideal 1 qubit quantum channel, we have C(S,T)=1, but C(T,S)=0, because quantum information cannot be sent on classical channels. On the other hand, with entanglement assistance we have C(S,T)=2 by superdense coding and C(T,S)=1/2 by teleportation.

Because all ideal channels S are equivalent as reference channels, we can define C_\text{E}(T)=C_\text{E}(T,S_1), with S_1 the ideal classical 1 bit channel as the entanglement assisted capacity of T. For this quantity there is an explicit formula (coding theorem) by [1]. The problem stated above first appeared in [2] as the “Reverse Shannon Theorem”.

Solution

The reversibility of entanglement-assisted coding was first shown in [3] by constructing a suitable simulation protocol. An alternative, independent proof appeared concurrently in [4]. The two proofs share similarities such as relying on the use of the “embezzling states” of [5], but they differ in their approach, with the former generalizing classical information-theoretic ideas through representation theory, and the latter employing one-shot information-theoretic methods.

References

[1] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett. 83, 3081 (1999), quant-ph/9904023.

[2] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem, quant-ph/0106052

[3] C. H. Bennett, I. Devetak, A. W. Harrow, P. W. Shor, and A. Winter, The Quantum Reverse Shannon Theorem and Resource Tradeoffs for Simulating Quantum Channels, IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2926–2959, 2014.

[4] M. Berta, M. Christandl, and R. Renner, The quantum reverse Shannon theorem based on one-shot information theory, Communications in Mathematical Physics, vol. 306, no. 3, pp. 579–615, 2011.

[5] W. van Dam and P. Hayden, Universal entanglement transformations without communication, Phys. Rev. A  67, 060302 (2003).